JMPJournal of Modern Physics2153-1196Scientific Research Publishing10.4236/jmp.2020.1111118JMP-104358ArticlesPhysics&Mathematics Calculations Energy of the (<i>nl</i><sup>2</sup>) <sup>1</sup><i>L<sup>&pi;</sup></i> Doubly Excited States of Two-Electron Systems via the Screening Constant by Unit Nuclear Charge Formalism MomarTalla Gning1*IbrahimaSakho1MalickSow2Department of Experimental Sciences, UFR Sciences and Technologies University of Thies, Thies, SenegalAtoms Laser Laboratory, Department of Physics, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal0411202011111891191011, July 202022, November 2020 25, November 2020© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this work, the total energies of doubly excited states ( ns 2) 1S e, ( np 2) 1D e, ( nd 2) 1G e, ( nf 2) 1I e, ( ng 2) 1K e, and ( nh 2) 1M e of the helium isoelectronic sequence with Z ≤ 10 are calculated in the framework of the variational method of the Screening Constant by Unit Nuclear Charge (SCUNC). These calculations are performed using a new wavefunction correlated to Hylleraas-type. The possibility of using the SCUNC method in the investigation of high-lying Doubly Excited States(DES) in two-electron systems is demonstrated in the present work in the case of the ( nl 2) 1 L &pi; doubly excited states, where accurate total energies are tabulated up to n = 20. All the results obtained in this paper are in agreement with the values of the available literature and may be useful for future experimental and theoretical studies on the doubly excited ( nl 2) 1 L π states of two-electron systems.

Doubly Excited States Helium-Like Systems Screening Constant by Unit Nuclear Charge (SCUNC) Wave Functions Correlated Total Energy1. Introduction

Studies of doubly-excited states of helium-like systems remain an active field of research since the early experiments of Madden and Codling [1] [2] concerning the observation of resonant structures in the absorption spectrum of helium using synchrotron radiation. The strong correlation between electrons in the doubly excited state of two-electron atomic systems has attracted considerable attention from theorists and experimenters as evidenced by the efforts concentrated in the field over the last twenty years. Theoretical investigations in two-electron Doubly Excited States (DES) are of great interest in connection with the understanding of collisional and radiative processes which take place in hot astrophysical and laboratory plasma [3] [4] [5]. In these investigations, great attention has been paid to the study of symmetric (nl2) DES where the electron-electron correlation effects may be predominant as revealed by the works of Fano [6]. Several experimental and theoretical studies on doubly excited (nl2) states have been carried out using different methods. Experimentally, many of these doubly excited states have been observed in electronic impact experiments by Oda et al. [7] and Hicks and Comer [8]. In their studies, these authors have worked on the energy spectra of ejected electrons from autoionization states in helium excited by electron impact. Other doubly excited states were observed by ion impact by Rudd [9] and by Bordenave-Montesquieu et al. [10]. These doubly excited states were also studied by examining the spectra of ejected electrons by Gelabart et al. [11] and by Rodbro et al. [12]. From a theoretical point of view, several calculation methods have been used, the complex rotation method [13] [14] [15], the variational method of Hylleraas [16], the double sums over the total hydrogen spectrum formalism [17], the density functional theory [18], the formalism of the Feshbach projection operators [19] [20], the discretization technique [21], the truncated diagonalization method [22], the time-dependent variation perturbation theory [23], and the semi-empirical procedure of the screening constant by unit nuclear charge (SCUNC) method [24] [25] [26], to name a few. In all these ab-initio methods, energies of (nl2) doubly-excited states of He isoelectronic sequence can’t be expressed in an analytical formula. In addition, most of these preceding methods require large basis-set calculations involving a fair amount of mathematics complexity. But, it’s widely believed that there are distinct advantages to viewing problems of physics within the framework of simple analytical models. Contrary to all these methods, the variational procedure of the SCUNC method makes it possible to calculate the energies of the (nl2) doubly excited states without a complex mathematical program or calculation code but from a simple analytical expression. In addition, in the recent past the variational procedure of the SCUNC method has been successfully applied to calculations of the energies of doubly excited states nlnl' (n = 2 - 4) in helium-like ions by Sakho [27] using a special Hylleraas-type wave function. These reasons sufficiently justify our choice to apply the variational procedure of the SCUNC method in this study. The goal of the present work is to report accurate total energies and excitation energies of the doubly excited states (nl2) using the variational procedure of the SCUNC formalism but also to show that it is possible to calculate total energies and precise excitation energies of high-lying up to n = 20 of the doubly excited states (nl2) without any calculation code or complex mathematical program, without a powerful computer but using a simple analytical expression. Section 2 gives the correlated wave functions and brief overview of the calculation method. Section 3 gives the presentation and the discussion of our results in the case of doubly excited states (ns2) 1Se, (np2) 1De, (nd2) 1Ge, (nf2) 1Ie, (ng2) 1Ke, and (nh2) 1Me of the He-like ions up to Z = 10 are made. All our results are compared to available theoretical and experimental data.

2. Theory and Calculations2.1. Hamiltonian and Hylleraas-Type Wave Functions

The time independent Schrödinger equation for the Helium atom, or the positive ions of its isoelectronic sequence, or of the negative Hydrogen ion, is

H ^ Ψ = E Ψ (1)

where H ^ represents the Hamiltonian operator of the considered system, Ψ the trial wave function and E the associated energy.

The Hamiltonian H of the helium isoelectronic series in given by (in atomic units)

H = − 1 2 Δ 1 − 1 2 Δ 2 − Z r 1 − Z r 2 + 1 r 12 (2)

In this equation, r1 and r2 denote the position of the two electrons from the nucleus, Z is the nuclear charge, Δ1 is the Laplacian with reference to the coordinates of the vector radius r1 which detect the position of the electron 1. Δ2 Laplacian defines the coordinates of the vector radius r2 which detect the position of the electron 2 and r12 inter-electronic distance.

Solving Equation (1) is very difficult because of the term r 12 = u = | r 1 − r 2 | representing electron-electron repulsion. It is therefore necessary to implement a rough calculation method using a correlated wavefunction.

The groundbreaking work in this area was conducted by Hylleraas [28] [29] [30]. The simplest Hylleraas wave function is written as follow:

Ψ ( r 1 , r 2 , r 12 ) = ( 1 + b r 12 ) exp [ − α ( r 1 + r 2 ) ] (3)

Since Hyllerass’ original work, tremendous efforts have been made to improve upon that work, using larger and larger expansions, adding more complicated terms. In this present work, we have modified this Hylleraas wavefunction in order to adapt it to the study of symmetrical (nl2) 1Lπ doubly excited states in two-electron atomic systems. These wave functions are defined as follow:

Ψ ( r 1 , r 2 , r 12 ) = ∑ υ = 0 υ = n − l − 1 ( n 2 r 0 2 ) υ × ( 1 + C 0 Z r 12 ) exp [ − α ( r 1 + r 2 ) ] (4)

In this expression, n is the principal quantum number; ℓ is orbital quantum number, r0 is Bohr radius, C0 and α are the variational parameters to be determined by minimizing the energy, Z is the nuclear charge number, r12 represents the term electron-electron repulsion r1 and r2 are the coordinates of electrons with respect to the nucleus.

From the theoretical viewpoint, the Hylleraas variational method is based on the Hylleraas and Undheim theorem [31] according to which, a good approximation of the energy eigenvalue E ( α , C 0 ) is obtained when the minima of the function ( d 2 E ( α , C 0 ) / d α d C 0 ) converge with increasing values of the dimension D of the basis states and when the function exhibit a plateau.

Using this theorem, the values of the varitional parameters α and C0 can be determined by the following conditions:

∂ E ( α , C 0 ) ∂ C 0 = 0 (5)

and

∂ E ( α , C 0 ) ∂ α = 0 (6)

In the framework of the Ritz’ variation principle, the energy E   ( α , C 0 ) = 〈 H 〉 ( α , C 0 ) is calculated from the relation:

E   ( α , C 0 ) = 〈 H 〉 ( α ) = 〈 Ψ ( α , C 0 ) | H | Ψ ( α , C 0 ) 〉 〈 Ψ ( α , C 0 ) | Ψ ( α , C 0 ) 〉 (7)

In this equation, the correlated wave functions are given by (4) and the Hamiltonian H of the helium isoelectronic series in given by (2) (in atomic units).

Furthermore, the closure relation represents the fact that | r 1 , r 2 〉 are continuous bases in the space of the two-electron space, written as follow:

∬ d r 1 3 d r 2 3 | r 1 , r 2 〉 〈 r 1 , r 2 | = 1 (8)

Using this relation, according to (7), we obtain:

E ( α , C 0 ) ∬ d r 1 3 d r 2 3 〈 Ψ ( α , C 0 ) | | r 1 , r 2 〉 × 〈 r 1 , r 2 | | Ψ ( α , C 0 ) 〉 = ∬ d r 1 3 d r 2 3 〈 Ψ ( α , C 0 ) | | r 1 , r 2 〉 H ^ 〈 r 1 , r 2 | | Ψ ( α , C 0 ) 〉 (9)

By developing this expression (9), we find:

E ( α , C 0 ) ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) × Ψ * ( α , C 0 ) = ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) H ^ Ψ * ( α , C 0 ) (10)

This means:

N * E ( α , C 0 ) = ∬ d r 1 3 d r 2 3 Ψ ( α , C 0 ) H ^ Ψ * ( α , C 0 ) (11)

with the normalization constant

N = ∬ d r 1 3 d r 2 3 | Ψ ( α , C 0 ) | 2 (12)

To make it easier to integrate Equation (11), we operate the variable changes in elliptic coordinates by:

s = r 1 + r 2 ;   t = r 1 − r 2 ;   u = r 12 (13)

On the basis of these variable changes, the elementary volume element

d τ = d 3 r 1 d 3 r 2 = 2 π 2 ( s 2 − t 2 ) u d s d u d t (14)

Using these elliptical coordinates, Equation (11) is written as follows

N E ( α , C 0 ) = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t { u ( s 2 − t 2 ) × [ ( ∂ Ψ ∂ s ) 2 + ( ∂ Ψ ∂ t ) 2 + ( ∂ Ψ ∂ u ) 2 ]   + 2 ( ∂ Ψ ∂ u ) × [ s ( u 2 − t 2 ) × ∂ Ψ ∂ s + t ( s 2 − u 2 ) × ∂ Ψ ∂ t − Ψ 2 × ( 4 Z s u − s 2 + t 2 ) ] } (15)

with respect to the correlated wave functions given by expression (4), it is expressed as follow

Ψ ( s , t , u , α , C 0 ) = ∑ υ = 0 υ = n − l − 1 ( n 2 r 0 2 ) υ × ( 1 + C 0 Z u ) exp ( − α s ) (16)

Furthermore, according to (12), the normalization constant is written in elliptic coordinates as:

N = ∫ 0 ∞ d s ∫ 0 s d u ∫ 0 u d t   u ( s 2 − t 2 ) × Ψ 2 (17)

2.2. General Formalism of the SCUNC Method

The Screening Constant by Unit Nuclear Charge (SCUNC) formalism is used in this work to calculate the total energies of the symmetrical (nl2) 1Lπ doubly excited states of the helium-isoelectronic up to Z = 10.

In the framework of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism, total energies of the (Nℓnℓ') 2S+1Lπ doubly excited states are expressed in Rydberg (Ry) as below [24] [25]

E ( N l n l ′ , L 2 s + 1 π ) = − Z 2   ( 1 N 2 + 1 n 2 [ 1 − β ( N l n l ′ , L 2 s + 1 π , Z ) ] 2 ) Ry (18)

In this equation, the principal quantum numbers N and n, are respectively the inner and the outer electron of the helium-isoelectronic series. In this equation, the β-parameters are screening constant by unit nuclear charge expanded in inverse powers of Z and given by

β   ( N l n l ′ , L 2 s + 1 π , Z ) = ∑ k   = 1 q f k ( 1 Z ) k (19)

where f k = f k ( N l n l ′ , L 2 s + 1 π ) are screening constants to be evaluated based on variational predicable using a wavefunction.

For the states (nl2) 1Lπ, N = n and l = l'. Hence, the total energy is written as follow:

E ( n l 2 , L 1 π ) = − Z 2 n 2 { 1 + [ 1 − β ( n l 2 , L 1 π , Z ) ]   2 } Ry (20)

Furthermore, in the framework of the screening constant by unit nuclear charge formalism, the β-screening constant is expressed in terms of the variational α-parameter as follow

β   ( n l 2 , L 1 π , Z , α ) = α Z 2 ( 1 + 2 L 2 n + 4 L − 3 ) (21)

In this expression, n denotes the principal quantum number, L characterizes the considered quantum state (S, P, D, F etc.) and α is the variational parameter.

Then, using Equation (21), the total energies of the symmetrical (nl2) 1Lπ doubly excited states in the helium isoelectronic series is expressed in Rydberg (Ry) as below:

E ( n l 2 , L 1 π , Z ) = − Z 2   n 2 { 1 + [ 1 − α Z 2 ( 1 + 2 L 2 n + 4 L − 3 ) ] 2 } Ry (22)

In this equation, only the parameter α is unknown. Considering the (2s2) 1Se state of Helium-like ions (Z = 2 - 10), we calculated the values of the variational parameters α and C0, the results of which are presented in Table 1.

The Equation (22) is used to calculate the total energies of the (nl2) 1Lπ doubly excited states of helium-like ions without a complex calculation program.

3. Results and Discussions

The results obtained in the present study for (ns2) 1Se, (np2) 1De, (nd2) 1Ge, (nf2) 1Ie, (ng2) 1Ke, and (nh2) 1Me with n ≤ 20 in the helium-like ions up to Z = 10 are listed in Tables 1-16 and compared to various other calculations. Table 1 presents our results on the calculation of the variational parameters α and C0. These variational parameters are calculated by determining the expression of E = f(a, C0) from the expression (15) and the wavefunction (16), then according to conditions (5) and (6) we obtained a system of equations whose resolution to give the values of the variational parameters α and C0 with 2 ≤ Z ≤ 10. All calculations in this work were performed with the calculation program MAXIMA. In Tables 2-7 we have listed our present results E on the calculation of the total energies of the (nl2) 1Lπ doubly excited states of the helium isoelectronic sequence with 2 ≤ Z ≤10 and 2 ≤ n ≤ 20 obtained using Equation (22). Table 2 shows our present results of the (ns2) 1Se (n = 2 - 20) doubly excited states of helium - like systems (Z = 2 - 10). Table 3 shows our present results of the (np2) 1De (n = 2 - 20) doubly excited states of helium-like systems (Z = 2 - 10). Table 4 shows our present results of the (nd2) 1Ge (n = 3 - 20) doubly excited states of helium-like systems (Z = 2 - 10). Table 5 shows our present results of the (nf2) 1Ie (n = 4 - 20) doubly excited states of helium-like systems (Z = 2 - 10). Table 6 shows our present results of the (ng2) 1Ke (n = 5 - 20) doubly excited states of helium-like systems (Z = 2 - 10). Table 7 shows our present results of the (nh2) 1Me (n = 6 - 20) doubly excited states of helium-like systems (Z = 2 - 10).

Table 8 shows a comparison of the present calculations for the (ns2) 1Se states with the results of the semi-empirical procedure of the screening constant by

Values of variational parameters α and C<sub>0</sub> of helium-like ions (Z = 2 - 10)
Z2345678910
α0.961051.403621.938372.473943.009973.546274.082764.619385.15608
C0 0.249820.265390.281390.291150.297710.302440.306000.308780.31102
 Total energy (-E) for (ns<sup>2</sup>) <sup>1</sup>S<sup>e</sup> (n = 2 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
ns2−E−E−E−E−E−E−E−E−E
2s221.4594052.4227696.46002154.07753225.28724310.09472408.50290520.51348646.12749
3s29.5375123.2990042.8711268.47890100.12766137.81987181.55684231.33932287.16777
4s25.3648513.1056924.1150138.5193856.3218177.52368102.12572130.12837161.53187
5s23.433508.3876415.4336024.6524136.0459649.6151565.3604683.28216103.38040
6s22.384385.8247510.7177817.1197325.0319234.4549745.3892157.8348371.79194
7s21.751794.279417.8742912.5777618.3908025.3138533.3471842.4909052.74510
8s21.341213.276426.028759.6298514.0804519.3809225.5314332.5320940.38297
9s21.059722.588784.763467.6087711.1253015.3133220.1729825.7043731.90753
10s20.858382.096913.858406.163109.0114912.4037916.3401220.8205425.84510
11s20.709401.732983.188765.093477.4475110.2510713.5042317.2070621.35959
12s20.596091.456192.679454.279936.257988.6137411.3473014.4587117.94799
13s20.507911.240782.283083.646815.332247.339529.6687112.3198515.29296
14s20.437951.069851.968573.144444.597706.328468.3367910.6227213.18628
15s20.381500.931961.714842.739164.005115.512797.262279.2535711.48671
16s20.335300.819111.507192.407463.520114.845236.382868.1330210.09574
17s20.297020.725571.335092.132563.118164.291975.654027.204348.94294
18s20.264930.647191.190861.902192.781323.828335.043256.426097.97688
19s20.237780.580861.068811.707232.496263.435954.526355.767467.15931
20s20.214590.524230.964601.540782.252873.100954.085035.205136.46127
 Total energy (-E) for the (np<sup>2</sup>) <sup>1</sup>D<sup>e</sup> (n = 2 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
np2−E−E−E−E−E−E−E−E−E
2p219.4069148.9877991.46705147.50214217.11646300.32082397.12093507.52007631.52025
3p28.7808922.0402041.0450666.0769897.14529134.25433177.40638226.60274281.84424
4p25.0013512.5033423.2424437.3725854.8986375.82286100.14645127.87008158.99418
5p23.230518.0522414.9482224.0148535.2550748.6702464.2610882.02799101.97124
6p22.259355.6186210.4197116.7283824.5466033.8752644.7148457.0656170.92775
7p21.669264.143597.6780112.3201518.0714124.9324132.9034941.9848652.17663
8p21.283863.182165.892609.4512013.8590019.1164825.2238832.1813539.98897
9p21.018242.520674.665127.4797710.9654215.1224219.9509825.4512131.62316
10p20.827392.046093.785056.066908.8922712.2614616.1746020.6318025.63311
11p20.685641.694053.132585.019817.3562310.1421013.3775217.0625821.19731
12p20.577481.425702.635464.222276.186548.5284611.2481414.3456417.82100
13p20.493051.216452.248003.600825.275277.271529.5896412.2297015.19171
14p20.425901.050141.940143.107184.551546.273378.2727410.5496913.10425
15p20.371590.915761.691482.708543.967185.467537.209659.1935711.41933
16p20.327060.805631.487762.382003.488574.807596.339108.0831310.03971
17p20.290080.714241.318752.111153.091654.260335.617237.162408.89585
18p20.259040.637571.177001.884022.758823.801485.012036.390517.93692
19p20.232730.572631.056941.691682.477003.412974.499635.737007.12510
20p20.210240.517120.954361.527362.236263.081124.061995.178866.43177
 Total energy (-E) for the (nd<sup>2</sup>) <sup>1</sup>G<sup>e</sup> (n = 3 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
nd2−E−E−E−E−E−E−E−E−E
3d28.6698721.8494440.7653265.7066796.68358133.70073176.76056225.86449281.01340
4d24.9202812.3651223.0402537.1053454.5657575.4239999.68137127.33865158.39628
5d23.172277.9535214.8041023.8245735.0182348.3866063.9304881.65034101.54646
6d22.216705.5466610.3148116.5900124.3744733.6691944.4747356.7913970.61936
7d21.637274.089807.5997012.2169317.9430624.7788132.7245741.7805651.94691
8d21.259313.141005.832749.3723513.7609918.9992225.0873132.0254339.81368
9d20.999012.488524.618397.4182510.8889715.0309919.8445125.3296731.48654
10d20.812062.020513.747906.018018.8315412.1888316.0900520.5352925.52462
11d20.673231.673373.102574.980327.3072010.0834713.3092816.9846921.10977
12d20.567281.408752.610884.189936.146398.4804711.1922814.2818917.74935
13d20.484581.202392.227613.574015.241997.231749.5433412.1768615.13234
14d20.418781.038341.923043.084704.523646.240038.2339410.5054213.05450
15d20.365550.905761.677012.689513.943575.439317.176819.1561111.37723
16d20.321890.797091.475392.365753.468414.783506.311068.0511510.00378
17d20.285630.706891.308102.097173.074304.239605.593117.134898.86493
18d20.255180.631191.167771.871902.743783.783514.991126.366667.91012
19d20.229360.567061.048881.681102.463883.397294.481395.716207.10173
20d20.207280.512230.947291.518072.224743.067374.045985.160616.41126
 Total energy (-E) for the (nf<sup>2</sup>) <sup>1</sup>I<sup>e</sup> (n = 4 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
nf2−E−E−E−E−E−E−E−E−E
4f24.8846212.3038122.9503236.9862654.4172875.2459599.47366127.10120158.12904
5f23.144697.9063214.7349723.7331434.9042848.2500163.7711981.46828101.34160
6f22.195225.5100510.2612716.5192424.2863233.5635844.3515956.6506970.46106
7f21.620304.060987.5575912.1613117.8738224.6958732.6278941.6701051.82266
8f21.245693.117955.799099.3279313.7057118.9330325.0101631.9373139.71456
9f20.987922.469804.591107.3822410.8441714.9773619.7820125.2582931.40626
10f20.802912.005113.725465.988428.7947412.1447816.0387220.4766825.45872
11f20.665601.660553.083904.955717.2766010.0468513.2666216.9359821.05500
12f20.560851.397962.595184.169256.120678.4496911.1564414.2409717.70334
13f20.479111.193222.214283.556455.220177.205639.5129412.1421515.09331
14f20.414081.030481.911633.069674.504966.217688.2079210.4757213.02111
15f20.361500.898981.667152.676543.927465.420047.154379.1305011.34844
16f20.318360.791191.466832.354483.454414.766766.291588.028919.97877
17f20.282530.701731.300622.087313.062064.224965.576087.115448.84307
18f20.252450.626651.161181.863232.733023.770644.976156.349567.89091
19f20.226940.563041.043061.673442.454363.385924.468155.701097.08475
20f20.205130.508660.942111.511272.216293.057264.034225.147196.39618
 Total energy (-E) for the (ng<sup>2</sup>) <sup>1</sup>K<sup>e</sup> (n = 5 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
ng2−E−E−E−E−E−E−E−E−E
5g23.128607.8786514.6943923.6794034.8372848.1696763.6774581.36112101.22099
6g22.182295.4878810.2287816.4762724.2327533.4993744.2766956.5650870.36472
7g21.609794.043017.5312912.1265417.8304924.6439532.5673441.6009151.74480
8g21.237043.103205.777539.2994313.6702218.8905024.9605831.8806539.65082
9g20.980712.457554.573207.3586010.8147414.9420919.7409125.2113331.35343
10g20.796851.994823.710445.968588.7700512.1152116.0042620.4373125.41443
11g20.660441.651823.071174.938917.2556910.0218113.2374316.9026421.01750
12g20.556431.390492.584284.154876.102798.4282911.1314914.2124817.67130
13g20.475291.186772.204883.544065.204757.187189.4914412.1176015.06570
14g20.410751.024881.903473.058914.491586.201678.1892610.4544112.99715
15g20.358580.894081.660022.667133.915765.406047.138079.1118811.32750
16g20.315790.786871.460552.346213.444134.754456.277248.012549.96037
17g20.280260.697911.295062.080003.052974.214085.563417.100978.82681
18g20.250420.623261.156241.856732.724943.760974.964896.336717.87646
19g20.225130.560011.038651.667632.447153.377294.458105.689627.07185
20g20.203500.505940.938161.506062.209823.049524.025215.136906.38462
 Total energy (-E) for the (nh<sup>2</sup>) <sup>1</sup>M<sup>e</sup> (n = 6 - 20) doubly excited states of helium-like systems (Z = 2 - 10). The energies E are in eV
Z2345678910
ng2−E−E−E−E−E−E−E−E−E
6h22.173655.4730210.2069816.4474024.1967533.4562044.2263356.5075070.29992
7h21.602634.030757.5133112.1027517.8008324.6083932.5258641.5535051.69145
8h21.231053.092965.762539.2796013.6455118.8608824.9260331.8411739.60639
9h20.975652.448914.560567.3418910.7939214.9171519.7118225.1780831.31603
10h20.792531.987463.699685.954368.7523412.0939915.9795220.4090425.38263
11h20.656721.645503.061934.926707.2404910.0036013.2162016.8783920.99022
12h20.553201.385012.576284.144316.089648.4125311.1131314.1915017.64770
13h20.472471.181992.197903.534855.193297.173459.4754412.0993215.04514
14h20.408271.020681.897343.050824.481526.189628.1752210.4383712.97912
15h20.356380.890371.654612.660003.906885.395427.125689.0977311.31160
16h20.313840.783581.455752.339873.436254.745026.266257.999999.94626
17h20.278510.694971.290782.074353.045944.205675.553617.089798.81423
18h20.248860.620621.152411.851672.718653.753444.956116.326697.86519
19h20.223720.557631.035201.663082.441493.370524.450215.680617.06172
20h20.202220.503800.935041.501952.204713.043414.018095.128776.37547
 Comparison of the present calculations on total energies of the doubly (ns<sup>2</sup>) <sup>1</sup>S<sup>e</sup> (n = 2 - 7) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States ns2 1Se Z2345678910
2s2 1Se−Ep 21.4594052.4227696.46002154.07753225.28724310.09472408.50290520.51348646.12749
−Ea 21.1900451.7567295.92923153.70728225.09117310.08075408.67588520.87683646.68334
−Eb 21.1669251.8606796.15032154.04035225.53336310.63042409.33154521.63766647.54894
−Ec 21.1938851.7571995.92188153.72003225.09205310.08931408.67561520.87758646.68514
−Ed 21.1667851.8605496.15046154.03995225.53363310.62879409.33289521.62923647.54853
−Ee 21.1646051.8631296.14556154.03886225.53540310.62852409.33358521.63576647.54377
−Ek21.1900052.0000096.43000154.45000226.09000311.32000410.17000522.62000648.67000
3s2 1Se−Ep9.5375123.2990042.8711268.47890100.12766137.81987181.55684231.33932287.16777
−Ea 9.6456223.3809743.1632468.99254100.86877138.79205182.76224232.77934288.84349
−Ec9.6406523.4806643.2942569.06061100.76559138.39575181.93676231.37273286.68564
−Ef 9.6203123.4058643.2334469.10874101.02769138.99573183.00878233.07091289.17806
−Eg 9.4177223.0030042.6351968.31431100.04059137.81366181.63365231.50082287.41489
−Ed 9.6201723.4058643.2331669.10847101.02837138.99627183.00918233.06996289.17887
−Ee 9.6246623.4017843.2252769.10738101.02224138.98212182.99653233.05866289.17125
−Eh 9.5838522.8385242.2565767.85978
−Ek9.4200023.1100042.8600068.65000100.48000138.37000182.30000232.27000288.30000
4s2 1Se−Ep5.3648513.1056924.1150138.5193856.3218177.52368102.12572130.12837161.53187
−Ea5.4913913.2597124.4294439.0005956.9733278.34732103.12276131.29961162.87803
−Ei5.4696313.2584824.4507939.0397857.0224378.41957103.20357131.40206162.99422
−Eg5.2975112.9391523.9823538.4268456.2727677.52023102.16899130.21932161.67093
−Ej5.3500013.1800024.4900038.0800057.9500078.43000103.18000130.42000163.20000
−Ek5.3000013.0000024.1100038.6100056.5200077.83000102.54000130.65000162.17000
−Ep3.433508.3876415.4336024.6524136.0459649.6151565.3604683.28216103.38040
5s2 1Se−Ea3.539938.5279215.6928125.0346236.5532050.2488466.1213884.17083104.39706
−Eg3.390408.2811115.3487224.5932536.0145549.6129165.3881683.34034103.46942
−Ej3.220008.1500015.3300024.2800036.4500049.6900065.2900083.20000103.67000
−Ek3.390008.3200015.4300024.7100036.1700049.8100065.6300083.62000103.79000
6s2 1Se−Ep2.384385.8247510.7177817.1197325.0319234.4549745.3892157.8348371.79194
−Ea2.470255.9416110.9248317.4196525.4260634.9446045.9744758.5164772.57007
−Ek2.350005.7800010.7100017.1600025.1200034.5900045.5700058.0700072.07000
7s2 1Se−Ep1.751794.279417.8742912.5777618.3908025.3138533.3471842.4909052.74510
−Ej1.720004.270007.8200012.6300018.4600025.3900033.4400042.5800052.83000
−Ek1.730004.240007.8700012.6100018.4600025.4100033.4800042.6600052.95000
−Ea1.8212594.375598.0404212.8160218.7023925.6995333.8071743.0255753.35447

pPresent work, values calculated from Equation (22), asakho [32], bHo [33], cSow et al. [36], dGning et al. [37], eKonte et al. [38], fHo [34], gSakho et al. [24], hRay et al. [23], iHo [35], jDiouf et al. [39], kSakho [40].

unit nuclear charge method of Sakho et al. [24] [32], with the results of the complex rotation of Ho [33] [34] [35], with the results of Sow et al. [36] who used the variational method of the SCUNC formalism, with the complex rotation values of Gning et al. [37], with the Konté et al. [38] data, with the data from the time-dependent variation perturbation theory of Ray et al. [23], with the results of Diouf et al. [39] and finally with the data from the modified slater theory of Sakho [40]. The observation of our results shown in this table shows that our present calculations are generally in good agreement with the results of the cited authors up to Z = 10.

In Table 9, we compare our calculations for (np2) 1De states with the results of the calculations of Badiane et al. [41], with the data of Sakho [24] [32] [40], with the results of the complex rotation calculations (CRC) of Ho and Bhatia [13], with the values of Ivanov and Safronova [17], with the results of the variational method calculations of Hylleraas de Biaye et al. [16] and finally with those obtained by Roy et al. [18] who applied the functional density theory (FDT). Here, the agreements between the calculations are considered good. Table 10 compares our results for the (nd2) 1Ge (n = 3 - 10) states with those obtained by Badiane et al. [41], sakho [32] [40], Bachau et al. [19], Biaye et al. [16], Ivanov and Safronova [17], Diouf et al. [39], Ray et al. [23] and Roy et al. [18]. As regards the (nd2) 1Ge-levels, comparison shows also a good agreement up to Z = 10. Table 11 shows the results of our present calculations of the total energies of the doubly excited (nf2) 1Ie (n = 4 - 10) states of helium-like systems up to Z = 10, which we compare with those obtained by Biaye et al. [16], Badiane et al. [41], Sakho et al. [32] [40], Ho [20], Diouf et al. [39], and Sow et al. [36]. A comparative

Comparison of the present calculations on total energies of the doubly (np<sup>2</sup>) <sup>1</sup>D<sup>e</sup> (n = 2 - 5) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States np2 1De Z2345678910
2p2 1De−Ep 19.4069148.9877991.46705147.50214217.11646300.32082397.12093507.52007631.52025
−Ea 19.1214548.1927490.86974147.15243217.04218300.53626397.63604508.34153632.65271
−Eb 19.1200848.1900290.86565147.14835217.03537300.52809397.62652508.33201632.64183
−Ec 19.1010448.1804990.81667147.03678216.85169300.26551397.28229507.90071632.12481
−Ed 19.4153348.3913890.97314147.16059216.95238300.35123397.35577507.96601632.18195
−Ee 19.1840348.3913891.14321147.47352217.39320300.90769398.02245508.73746633.05408
−Ef47.7804990.54592146.90888
−Eg19.1200048.1900090.87000147.15000217.03000300.53000397.63000508.33000632.64000
−Eh19.1200048.7300091.95000148.77000219.20000303.24000400.88000512.13000636.9800
3p2 1De−Ep8.7808922.0402041.0450666.0769897.14529134.25433177.40638226.60274281.84424
−Ea8.9865622.5677742.2334567.9618299.74201137.57401181.45511231.38394287.36187
−Eb 9.1906522.6221942.1001167.6244099.19642136.81482180.48094230.19344285.95368
−Ec 9.3375922.9160842.5450268.2203399.94474137.71687181.53675231.40299287.31561
−Ed 8.9892922.1119841.2824166.4978597.76238135.07193178.43056227.83422283.28559
−Ee 8.6695521.8058541.0320666.3767697.83993135.44200179.18160229.06009285.07883
−Ef 21.8262641.0946566.40669
−Eg8.5000021.4200040.3800065.4000096.46000133.57000176.72000225.92000281.17000
−Eh8.5000021.6600040.8700066.1200097.42000134.77000178.17000227.61000283.10000
4p2 1De−Ep5.0013512.5033423.2424437.3725854.8986375.82286100.14645127.87008158.99418
−Ea5.1130212.8288123.9718738.5245356.4826977.84364102.61009130.77661162.34727
−Eb 5.38377413.0818724.1814138.6823656.5860977.88989102.59512130.70314162.21121
−Ec 13.1267824.2616838.7993756.7398478.08038102.82642130.97117162.51598
−Ee4.89396912.2356022.9392137.0224654.4935475.3606099.629084127.30035158.37441
−Ef12.3798223.2616637.543563
−Eg4.7800012.0500022.7200036.7900054.2600075.1300099.41000127.08000158.16000
−Eh4.7800012.1800022.9900037.1900054.8000075.81000100.22000128.03000159.24000
5p2 1De−Ep3.230518.0522414.9482224.0148535.2550748.6702464.2610882.02799101.97124
−Ea3.288498.2477715.4002924.7365236.2523849.9465265.8189283.86824104.09583
−Eb3.530688.5130915.6724025.0086336.5217850.2104766.0774384.12131104.34209
−Ef7.9511714.9295324.08344
−Eg7.7100014.5400023.5400034.7200048.0800063.6200081.33000101.22000
−Eh3.060007.8000014.7100023.8000035.0700048.5200064.1400081.94000101.92000

pPresent work, values calculated from Equation (22), aBadiane et al. [41], bSakho et al. [32], cHo and Bhatia [13], dIvanov and Safronova [17], eBiaye et al. [16], fRoy et al. [18], gSakho et al. [24], hSakho [40].

Comparison of the present calculations on total energies of the doubly (nd<sup>2</sup>) <sup>1</sup>G<sup>e</sup> (n = 3 - 10) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States nd2 1Ge Z2345678910
3d2 1Ge−Ep 8.6698721.8494440.7653265.7066796.68358133.70073176.76056225.86449281.01340
−Ea 8.3171620.8779439.5640164.3427195.23988132.11133175.08356224.11306279.19300
−Eb 8.3049221.0711439.8837464.7427195.64942132.60385175.60466224.65184279.74676
−Ec 8.3892721.1405339.8919164.6814895.48479132.35623175.26860224.22190279.18892
−Ed 8.5838321.5473440.4946465.4515796.431745133.44332176.491754225.57839280.70732
−Ee 8.5770321.2493839.9708264.7372795.551456132.41065175.321664224.27632279.27872
−Ef8.5000021.7300041.0600065.5200096.77000134.03000178.740000227.96000282.92000
−Eg8.4899621.1541439.8402164.56448
−Ei8.3500021.4300040.5500065.7200096.94000134.21000177.52000226.87000283.28000
4d2 1Ge−Ep4.9202812.3651223.0402537.1053454.5657575.4239999.68137127.33865158.39628
−Ea4.7810412.0301622.7432836.8904854.4581675.4395199.82773127.62417158.82476
−Eb 4.8531512.1770922.9038337.0306354.5588475.4898599.82092127.55342158.68869
−Ed 4.8939712.2356022.9378537.0211054.4921875.3578799.62500127.29354158.77713
−Eg 5.3116612.5172423.3310537.55445
−Ei4.7000012.0500022.8100036.9700054.5300075.4900099.85000127.62000158.78000
5d2 1Ge−Ep3.172277.9535214.8041023.8245735.0182348.3866063.9304881.65034101.54646
−Ea3.093947.7919814.7145623.8371835.1489648.6458164.3250282.18386102.22233
−Eb 3.179657.9225914.8424523.9378635.2115548.6621464.2896482.09406102.07538
−Eh 3.066727.8137514.7376923.83854
−Ef3.040007.5900014.5700023.60000034.8000048.3100063.8700081.59000101.68000
−Ei3.010007.7100014.6000023.6600034.9000048.3100063.9100081.67000101.62000
6d2 1Ge−Ep2.216705.5466610.3148116.5900124.3744733.6691944.4747356.7913970.61936
−Ea2.163315.4504410.2859116.6492924.5337933.9353344.8525457.2827171.22855
−Ef2.040005.3400010.1200024.2800033.5100044.3100056.7400070.50000
−Eh5.4599710.2818216.61392
−Ei 2.090005.3600010.1400016.4300024.2300033.5500044.3800056.7200070.57000
7d2 1Ge−Ep1.637274.089807.5997012.2169317.9430624.7788132.7245741.7805651.94691
−Ea1.595954.024567.5892612.2791418.0846925.0072733.0427942.1899152.44861
−Ef1.500003.930007.4400012.0400017.8700024.5700032.6400041.6800051.83000
−Eh3.874907.5729312.23152
−Ei1.530003.940007.4500012.0700017.8000024.6500032.6000041.6700051.85000
8d 21Ge−Ep1.259313..141005.832749.3723513.7609918.9992225.0873132.0254339.81368
−Ea1.225873.092575.828679.4273813.8805219.1881025.3474032.3584140.22114
9d2 1Ge−Ep0.999012.488524.618397.4182510.8889715.0309919,8445125.3296731.48654
−Ea0.971452.449024.616417.4640810.9865915.1853120.0561425.6004631.81826
10d2 1Ge−Ep0.812062.020513.747906.018018.8315412.1888316.0900520.5352925.52462
−Ea0.787771.987793.745656.055898.9117312.3158716.2642420.7582025.79638

pPresent work, values calculated from Equation (22), aBadiane et al. [41], bSakho et al. [32], cBachau et al. [19], dBiaye et al. [16], eIvanov and Safronova [17], fDiouf et al. [39], gRay et al. [23], hRoy et al. [18], iSakho [40].

Comparison of the present calculations on total energies of the doubly (nf<sup>2</sup>) <sup>1</sup>I<sup>e</sup> (n = 4 - 10) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States nf2 1Ie Z2345678910
4f2 1Ie−Ep 4.8846212.3038122.9503236.9862654.4172875.2459599.47366127.10120158.12904
−Ea 4.8858012.2151822.8956536.9462554.3737775.1877699.38820126.98190157.96206
−Eb 4.3442911.1158521.4942735.3298952.5982473.2884197.39225124.90568155.82597
−Ec 4.4055211.3553021.7078835.4605152.6145673.1714097.12829124.48662155.24772
−Ed 4.63682
−Ee 5.0300012.1200023.4300037.5400055.2000076.38000100.82000128.65000159.25000
−Ef5.0213912.4341123.2789237.6663555.3895976.43250100.78073128.41943159.33212
−Eg4.6200011.9200022.6300036.7400054.2500075.1600099.47000127.18000158.30000
5f2 1Ie−Ep3.144697.9063214.7349723.7331434.9042848.2500163.7711981.46828101.34160
−Eb2.806857.3552414.1635223.1895434.4142347.8294463.4297381.21236101.17735
−Ee 2.930007.5900014.4100023.5000034.6900048.1700063.5500081.41000100.29000
−Eg 2.950007.6300014.4800023.5100034.7200048.1000063.6600081.40000101.31000
6f2 1Ie−Ep2.195225.5100510.2612716.5192424.2863233.5635844.3515956.6506970.46106
−Eb1.976915.182419.9607216.2792124.1269733.4944844.3804056.7806270.69789
−Ee 2.070005.2800010.0400016.2200024.1000033.4100044.2200056.5400070.38000
−Eg2.050005.3000010.0600026.3300024.1100033.4000044.2100056.5300070.35000
7f2 1Ie−Ep1.620304.060987.5575912.1613117.8738224.6958732.6278941.6701051.82266
−Eb1.465333.838177.3688412.0328717.8193724.7228932.7420941.8742352.11931
−Ee1.500003.880007.3600011.9500017.7500024.5700032.4800041.5100051.63000
−Eg1.510003.890007.3900011.9900017.7100024.5400032.4800041.5300051.69000
8f2 1Ie−Ep1.245693.117955.799099.3279313.7057118.9330325.0101631.9373139.71456
−Eb1.127912.953795.666779.2069713.6873218.9853725.13515232.1366339.99120
9f2 1Ie−Ep0.987922.469804.591107.3822410.8441714.9773619.7820125.2582931.40626
−Eb0.893892.342894.491247.3266610.8410115.0315619.8956025.4344731.64547
10f2 1Ie−Ep0.802912.005113.725465.988428.7947412.1447816.0387220.4766825.45872
−Eb0.726541.902083.646325.945688.79607812.1934216.1377120.6275825.66169

pPresent work, values calculated from Equation (22), aBiaye et al. [16], bBadiane et al. [41], cSakho et al. [32], dHo [20], eDiouf et al. [39], fSow et al. [36]; gSakho [40].

reading of the results mentioned in this table shows a good agreement between the present SCUNC results and the data found in the literature. For level 4f2, it should be noted that comparison with the results of Biaye et al. [16] indicates satisfactory agreement for Z = 2 - 10.

In Table 12 and Table 13, we compare the results of our calculations of the total energies of the doubly excited states (ng2) 1Ke and (nh2) 1Me with those of Sakho [40] and Diouf et al. [39]. The agreements between the calculations are seen to be generally good. It is worth mentioning that there are not many results

Comparison of the present calculations on total energies of the doubly (ng<sup>2</sup>) <sup>1</sup>K<sup>e</sup> (n = 5 - 10) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States ng2 1Ke Z2345678910
5g2 1Ke−Ep 3.128607.8786514.6943923.6794034.8372848.1696763.6774581.36112101.22099
−Ea 2.930007.5900014.4200023.4400034.6300047.9900063.5400081.26000101.16000
−Eb 2.930007.5900014.4100023.4100034.6900047.9000063.7100081.24000101.28000
6g2 1Ke−Ep2.182295.4878810.2287816.4762724.2327533.4993744.2766956.5650870.36472
−Ea2.030005.2700010.0200016.2700024.0500033.3300044.1200056.4300070.25000
−Eb 2.040005.2300010.0400016.2200024.1000033.4100044.2200056.5400070.38000
7g2 1Ke−Ep1.609794.043017.5312912.1265417.8304924.6439532.5673441.6009151.74480
−Ea1.490003.870007.3600011.9600017.6700024.4900032.4200041.4600051.61000
−Eb1.500003.880007.3600011.9500017.7500024.4300032.4800041.5100051.63000
8g2 1Ke−Ep1.237043.103205.777539.2994313.6702218.8905024.9605831.8806539.65082
9g2 1Ke−Ep0.980712.457554.573207.3586010.8147414.9420919.7409125.2113331.35343
10g2 1Ke−Ep0.796851.994823.710445.968588.7700512.1152116.0042620.4373125.41443

pPresent work, values calculated from Equation (22), asakho [40], bDiouf et al. [39].

Comparison of the present calculations on total energies of the doubly (nh<sup>2</sup>) <sup>1</sup>M<sup>e</sup> (n = 6 - 10) excited states of helium-like systems (Z = 2 - 10) with available literature values. All results are expressed in eV
States nh2 1Me Z2345678910
6h2 1Me−Ep 2.173655.4730210.2069816.4474024.1967533.4562044.2263356.5075070.29992
−Ea 2.020005.250009.9900016.2400024.0100033.2800044.0700056.3700070.18000
−Eb 2.040005.230009.9700016.2200024.1000033.2700044.0600056.3700070.18000
7h2 1Me−Ep1.602634.030757.5133112.1027517.8008324.6083932.5258641.5535051.69145
−Ea1.480003.860007.3400011.9300017.6400024.4500032.3800041.4200051.56000
−Eb 1.460003.820007.3600011.9500017.6300024.4300032.3300041.5100051.63000
8h2 1Me −Ep1.231053.092965.762539.2796013.6455118.8608824.9260331.8411739.60639
9h2 1Me−Ep0.975652.448914.560567.3418910.7939214.9171519.7118225.1780831.31603
10h2 1Me−Ep0.792531.987463.699685.954368.7523412.0939915.9795220.4090425.38263

pPresent work, values calculated from Equation (22), aSakho [40], bDiouf et al. [39].

on the states and the only ones available to our knowledge are those of the authors Sakho [40] and Diouf et al. [39]. Moreover for n > 7 there are no results available so we think that the results cited up to n = 20 in this work may be interesting for future experimental and theoretical studies on these states.

In Tables 14-16 are quoted results for excitation energies of helium-like ions with Z ≤ 10. Our excitation energies are calculated with respect to the accurate ground state energies of Frankowski and Pekeris [42]. Comparison indicates that our excitation energies for (ns2) 1Se, (np2) 1De and (nd2) 1Ge levels agree well

Comparison of the present calculations on the variationnal calculation of the excitation energies of the doubly excited states (ns<sup>2</sup>) <sup>1</sup>S<sup>e</sup> (n = 2 - 5) of the He-like systems with some theoretical results available in the literature consulted for Z = 2 - 5. All the results are expressed in eV: 1 Ryd = 13.605698 eV
States ns2 1SeZ2345
2s2 1Se Ep57.5548145.6737275.1269445.4159
Ea57.8487146.2476275.4882445.5893
Eb58.6460147.7633277.6814448.4165
Ec57.8200146.3400275.6600445.7800
3s2 1SeEp69.4768174.7975328.7157531.0145
Eb69.4299175.2577329.3314531.6345
Ec69.5900175.1000328.9600531.1800
Ed69.3972174.7026328.4035530.5215
4s2 1SeEp73.6494184.9908347.4719560.9740
Ea73.5469184.8470347.1820560.5819
Eb73.4218184.9939347.5358561.0989
Ec73.7100185.1600347.6100561.0600
5s2 1SeEp75.5808189.7089356.1533574.8410
Eb75.1905189.4457355.9523574.7972
Ec75.6200189.8200356.2400574.9000

pPresent work, aHo [33], bRay and Mukherjee [23], cSakho et al. [24], dHo [34].

Comparison of the present calculations on the variationnal calculation of the excitation energies of the doubly excited states (np<sup>2</sup>) <sup>1</sup>D<sup>e</sup> (n = 2 - 5) of the He-like systems with some theoretical results available in the literature consulted for Z = 2 - 5. All the results are expressed in eV: 1 Ryd = 13.605698 eV
States np2 1DeZ2345
2p2 1De Ep59.6074149.1087280.1198451.9913
Ea59.8900149.9100280.7200452.3400
Eb 149.9157280.7699452.4575
Ec60.3249150.6749281.6815453.8316
Ed150.3239281.0420452.5854
Ee150.0599280.9522452.6670
3p2 1DeEp70.2334176.0563330.5418533.4165
Ea70.5100176.6900331.2100534.0900
Eb176.4251330.8008533.5502
Ec70.0693175.9788330.1368533.3216
Ed176.2699330.4933533.0876
Ef176.4441330.8253533.5611
4p2 1DeEp74.0129185.5932348.3444562.1209
Ea74.2300186.0500348.8700562.7000
Ec73.5851185.3858347.9848561.5942
Ed185.7178348.3249561.9479
5p2 1DeEp75.7838190.0443356.6386575.4786
Ea75.9500190.3900357.2000575.9500
Ec189.6852358.2598575.0067
Ed 190.1451356.6570575.4095

pPresent work, aSakho et al. [24]; bHo and Bhatia [13]; cRay and Mukherjee [23]; dRoy et al. [18]; eMacias et al. [21]; fBachau et al. [19].

Comparison of the present calculations on the variationnal calculation of the excitation energies of the doubly excited states (nd<sup>2</sup>) <sup>1</sup>G<sup>e</sup> (n = 2 - 5) of the He-like systems with some theoretical results available in the literature consulted for Z = 2 - 5. All the results are expressed in eV
States nd2 1GeZ2345
3d2 1Ge Ep70.3444176.2471330.8215533.7867
Ea70.6245176.9557331.6960534.8128
Eb 71.0544177.3529332.0144535.0495
Ec70.5238176.9421331.7477534.9298
4d2 1GeEp74.0939185.7314348.5466562.3881
Eb74.3089186.0362348.7685562.5167
Ec73.7021185.5789348.2569561.9398
5d2 1GeEp75.8420190.1429356.7827575.6688
Eb75.9470190.2838356.8475575.6543
Ec75.6667189.8185358.3822575.0611
6d2 1GeEp76.7975192.5498361.2720582.9034
Eb76.8667192.6376361.3047582.8789
7d2 1GeEP77.3770194.0067363.9872587.2765
Eb77.4273194.0717364.0150587.2600

pPresent work; aBachau et al. [19]; bRoy et al. [18]; cRay and Mukherjee [23].

with results obtained by the authors cited in Tables 14-16. Overall, we find a good agreement between our results and those of these authors. For all the states studied, the results obtained are in good agreement with the theoretical results we have consulted. The small difference noted between our results and those of the authors mentioned above is explained by the used method and the choice of the correlated wavefunction. The actual results presented in these different tables sufficiently show the good agreements between the current calculations and the different ab-initio results for the doubly excited states (ns2) 1Se, (np2) 1De, (nd2) 1Ge, (nf2) 1Ie, (ng2) 1Ke, and (nh2) 1Me of the He-like ions up to Z = 10. This very good agreement sufficiently justifies the validity of the variational procedure of the SCUNC method to give the precise values obtained directly from an analytical expression, unlike all the ab-initio methods cited in this document. Furthermore, the results quoted up to n = 20 in this work may be interesting for future experimental and theoretical studies in the doubly excited states (nl2) 1Lπ. In summary, the manuscript reports on new calculations for key atomic-structure parameters of important fundamental few-body systems (helium and helium-like ions). While not allowing more precision tests of physics due to the neglect of relativistic, spin, and QED effects, such results can still be helpful in the future development of theories to describe more complex atoms, or may be further developed to study the time-dependent evolution of atoms in external (e.g. laser) fields.

4. Conclusion

In this paper, the total energies and excitation energies of the doubly excited (ns2) 1Se, (np2) 1De, (nd2) 1Ge, (nf2) 1Ie, (ng2) 1Ke, and (nh2) 1Me states of helium-like ions up to Z =10 are reported. These energies are calculated in the framework of the variationnal procedure of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism. In this present work, a new wavefunction correlated to Hylleraas-type adapted to the correct description of electron-electron correlation phenomena in the (nl2) doubly excited states of helium-like systems has been constructed. Our results for total energies and the excitation energies are in good agreement with the values cited in the experimental and theoretical literature. Furthermore, for n > 10, no theoretical and experimental values from the literature are available for direct comparison. The good precision obtained in this work underlines that the results quoted up to n = 20 in this work may be interesting for future experimental and theoretical studies in the doubly excited states (nl2) 1Lπ. The results presented in this paper show that it is therefore possible to perform an analytical calculation of the total energies of the (nl2) doubly excited states for helium-like ions, without having to resort to excessively complicated calculations or a computer program.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Gning, M.T., Sakho, I. and Sow, M. (2020) Calculations Energy of the (nl2) 1Lπ Doubly Excited States of Two-Electron Systems via the Screening Constant by Unit Nuclear Charge Formalism. Journal of Modern Physics, 11, 1891-1910. https://doi.org/10.4236/jmp.2020.1111118

ReferencesMadden, R.P. and Codling, K. (1963) Physical Review Letters, 10, 516. https://doi.org/10.1103/PhysRevLett.10.516Codling, K., Madden, R.P. and Ederer, D.L. (1967) Physical Review Journals Archive, 155, 26. https://doi.org/10.1103/PhysRev.155.26Kahn, M.S. (1999) Physica Scripta T, 80, 23-27. https://doi.org/10.1238/Physica.Topical.080a00023Keller, S. and Dreizler, R.M. (1999) Physica Scripta T, 80, 93-94. https://doi.org/10.1238/Physica.Topical.080a00093Faenov, A.Ya., Skobelev, I.Yu. and Rosmej, F.B. (1999) Physica Scripta T, 80, 43-45. https://doi.org/10.1238/Physica.Topical.080a00043Fano, U. (1983) Reports on Progress in Physics, 46, 97-165. https://doi.org/10.1088/0034-4885/46/2/001Oda, N., Nishimura, F. and Tashira, S. (1970) Physical Review Letters, 24, 42-45. https://doi.org/10.1103/PhysRevLett.24.42Hicks, P.J. and Comer, J. (1975) Journal of Physics B: Atomic and Molecular Physics, 8, 1866-1879. https://doi.org/10.1088/0022-3700/8/11/022Rudd, M.E. (1965) Physical Review Letters, 15, 580-581. https://doi.org/10.1103/PhysRevLett.15.580Bordenave-Montesquieu, A., Gleizes, A., Rodiere, M. and Benoit-Cattin, P. (1973) Journal of Physics B: Atomic and Molecular Physics, 6, 1997. https://doi.org/10.1088/0022-3700/6/10/016Gelebart, F., Tweed, R.J. and Peresse, J. (1976) Journal of Physics B: Atomic and Molecular Physics, 9, 1739. https://doi.org/10.1088/0022-3700/9/10/018Rodbro, M., Bruch, R. and Bisgaard, P. (1979) Journal of Physics B: Atomic and Molecular Physics, 12, 2413-2447. https://doi.org/10.1088/0022-3700/12/15/009Ho, Y.K. and Bhatia, A.K. (1991) Physical Review A, 44, 2895-2855. https://doi.org/10.1103/PhysRevA.44.2895Lindroth, E. (1994) Physical Review A, 49, 4473-4480. https://doi.org/10.1103/PhysRevA.49.4473Ho, Y.K. and Callaway, J. (1985) Journal of Physics B: Atomic and Molecular Physics, 18, 3481. https://doi.org/10.1088/0022-3700/18/17/010Biaye, M., Konté, A., Faye, B.A.N. and Wagué, A. (2005) Physica Scripta, 71, 39-42. https://doi.org/10.1088/0031-8949/71/1/006Ivanov, A.I. and Safronova, I.U. (1993) Optika.Spectrosk, 75, 516.Roy, K.A., Singh, R. and Deb, M.B. (1997) Journal of Physics B: Atomic, Molecular and Optical Physics, 30, 4763. https://doi.org/10.1088/0953-4075/30/21/014Bachau, H., Martin, F., Riera, A. and Yanez, M. (1991) Atomic Data Nuclear Data Tables, 48, 167-212. https://doi.org/10.1016/0092-640X(91)90006-PHo, Y.K. (1989) Z Physics D Atoms, Molecules and Clusters, 11, 277-281. https://doi.org/10.1007/BF01438499Macias, A., Martin, F., Riera, A. and Yanez, M. (1987) Physical Review A, 36, 4187-4202. https://doi.org/10.1103/PhysRevA.36.4187Conneely, M.J. and Lipsky, L. (1978) Journal of Physics B: Atomic and Molecular Physics, 24, 4135. https://doi.org/10.1088/0022-3700/11/24/008Ray, D. and Mukherjee, P.K. (1991) Journal of Physics B: Atomic, Molecular and Optical Physics, 24, 1241. https://doi.org/10.1088/0953-4075/24/6/013Sakho, I., Ndao, A.S., Biaye, M. and Wagué, A. (2008) European Physical Journal D, 47, 37-44. https://doi.org/10.1140/epjd/e2008-00018-2Sakho, I., Ndao, A.S., Biaye, M. and Wagué, A. (2008) Physica Scripta, 77, Article ID: 055303. https://doi.org/10.1088/0031-8949/77/05/055303Sakho, I. (2011) European Physical Journal D, 61, 267-283. https://doi.org/10.1140/epjd/e2010-10294-8Sakho, I. (2011) Journal of Atomic and Molecular Sciences, 2, 20-42. https://doi.org/10.4208/jams.062910.072810aHylleraas, E.A. (1928) Zeitschrift für Physik, 48, 469-494. https://doi.org/10.1007/BF01340013Hylleraas, E.A. (1929) Zeitschrift für Physik, 54, 347-366. https://doi.org/10.1007/BF01375457Hylleraas, E.A. (1930) Zeitschrift für Physik, 65, 209-225. https://doi.org/10.1007/BF01397032Hylleraas, E.A. and Undheim, B. (1930) Zeitschrift für Physik, 65, 759-772. https://doi.org/10.1007/BF01397263Sakho, I., Ndao, A.S., Biaye, M. and Wagué, A. (2010) Physica Scripta, 82, Article ID: 035301. https://doi.org/10.1088/0031-8949/82/03/035301Ho, Y.K. (1981) Physical Review A, 23, 2137. https://doi.org/10.1103/PhysRevA.23.2137Ho, Y.K. (1979) Journal of Physics B: Atomic and Molecular Physics, 12, 387. https://doi.org/10.1088/0022-3700/12/3/016Ho, Y.K. (1980) Physics Letters A, 79, 44-46. https://doi.org/10.1016/0375-9601(80)90313-8Sow, B., Sow, M., Gning, Y., Traore, A., Ndao, A.S. and Wague, A. (2016) Radiation Physics and Chemistry, 123, 25-30. https://doi.org/10.1016/j.radphyschem.2016.01.037Gning, et al. (2015) Radiation Physics and Chemistry, 106, 1-6. https://doi.org/10.1016/j.radphyschem.2014.06.015Konté, A., Ndao, A.S., Biaye, M. and Wagué, A. (2006) Physica Scripta, 74, 605. https://doi.org/10.1088/0031-8949/74/6/002Diouf, A., et al. (2017) International Journal of Applied Mathematics and Theoretical Physics, 3, 78-85. https://doi.org/10.11648/j.ijamtp.20170304.11Sakho, I. (2010) Journal of Atomic and Molecular Sciences, 1, 103-117. https://doi.org/10.4208/jams.022510.031010aBadiane, J., et al. (2018) International Journal of Applied Mathematics and Theoretical Physics, 4, 27-41. https://doi.org/10.11648/j.ijamtp.20180402.11Frankowski, F. and Pekeris, L.C. (1966) Physical Review, 146, 46-49. https://doi.org/10.1103/PhysRev.146.46